Mod 15 (Wk 9)
Julianna D.
2023-11-02
Module 15 Multiple Regression and ANCOVA
R = matrix(cbind(1, 0.8, -0.5, 0, 0.8, 1, -0.3, 0.3, -0.5, -0.3, 1, 0.6, 0,
0.3, 0.6, 1), nrow = 4)n <- 1000
k <- 4
M <- NULL
V <- NULL
mu <- c(15, 40, 5, 23) # vector of variable means
s <- c(5, 20, 4, 15) # vector of variable SDs
for (i in 1:k) {
V <- rnorm(n, mu[i], s[i])
M <- cbind(M, V)
}
M <- matrix(M, nrow = n, ncol = k)
orig <- as.data.frame(M)
names(orig) = c("Y", "X1", "X2", "X3")
head(orig)## Y X1 X2 X3
## 1 10.789676 7.9721672 5.2036591 13.872677
## 2 8.932434 75.7880384 7.4068599 29.533060
## 3 15.013207 51.2004587 10.4151323 6.756778
## 4 19.121711 48.8518779 4.8960257 28.101257
## 5 15.177670 -0.8254089 7.3178741 8.504210
## 6 15.828277 75.7393465 0.3584238 -15.735571cor(orig)## Y X1 X2 X3
## Y 1.00000000 0.02007344 -0.00190345 -0.03394157
## X1 0.02007344 1.00000000 -0.04810534 -0.02557318
## X2 -0.00190345 -0.04810534 1.00000000 0.06401337
## X3 -0.03394157 -0.02557318 0.06401337 1.00000000plot(orig)
ms <- apply(orig, 2, FUN = "mean")
ms## Y X1 X2 X3
## 15.123685 39.690855 5.031446 23.183481sds <- apply(orig, 2, FUN = "sd")
sds## Y X1 X2 X3
## 5.018583 20.307670 3.916493 14.716197normalized <- sweep(orig, 2, STATS = ms, FUN = "-") # 2nd dimension is columns, removing array of means, function = subtract
normalized <- sweep(normalized, 2, STATS = sds, FUN = "/") # 2nd dimension is columns, scaling by array of sds, function = divide
head(normalized) # now a dataframe of Z scores## Y X1 X2 X3
## 1 -0.86359212 -1.5619068 0.04397113 -0.6326909
## 2 -1.23366504 1.7775148 0.60651535 0.4314688
## 3 -0.02201366 0.5667614 1.37461886 -1.1162329
## 4 0.79664445 0.4511115 -0.03457706 0.3341744
## 5 0.01075713 -1.9951213 0.58379458 -0.9974908
## 6 0.14039667 1.7751171 -1.19316494 -2.6446405M <- as.matrix(normalized)U = chol(R)
newM = M %*% U
new = as.data.frame(newM)
names(new) = c("Y", "X1", "X2", "X3")
cor(new)## Y X1 X2 X3
## Y 1.00000000 0.8043311 -0.5021422 -0.01203612
## X1 0.80433111 1.0000000 -0.3285746 0.26045570
## X2 -0.50214220 -0.3285746 1.0000000 0.61546733
## X3 -0.01203612 0.2604557 0.6154673 1.00000000plot(orig)
plot(new)
df <- sweep(new, 2, STATS = sds, FUN = "*") # scale back out to original mean...
df <- sweep(df, 2, STATS = ms, FUN = "+") # and standard deviation
head(df)## Y X1 X2 X3
## 1 10.789676 6.629608 5.849393 6.341751
## 2 8.932434 41.306875 10.626251 45.605142
## 3 15.013207 46.238980 10.019761 29.521066
## 4 19.121711 58.129863 3.650797 29.226502
## 5 15.177670 15.555859 5.651158 4.672821
## 6 15.828277 63.600853 1.943915 1.566435cor(df)## Y X1 X2 X3
## Y 1.00000000 0.8043311 -0.5021422 -0.01203612
## X1 0.80433111 1.0000000 -0.3285746 0.26045570
## X2 -0.50214220 -0.3285746 1.0000000 0.61546733
## X3 -0.01203612 0.2604557 0.6154673 1.00000000plot(df)
Challenge 1
library(ggplot2)
require(gridExtra)g1 <- ggplot(data = df, aes(x = X1, y = Y)) + geom_point() + geom_smooth(method = "lm",
formula = y ~ x)
g2 <- ggplot(data = df, aes(x = X2, y = Y)) + geom_point() + geom_smooth(method = "lm",
formula = y ~ x)
g3 <- ggplot(data = df, aes(x = X3, y = Y)) + geom_point() + geom_smooth(method = "lm",
formula = y ~ x)
grid.arrange(g1, g2, g3, ncol = 3)
m1 <- lm(data = df, formula = Y ~ X1)
summary(m1)##
## Call:
## lm(formula = Y ~ X1, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.655 -2.032 -0.072 2.008 8.084
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.309177 0.205652 35.54 <2e-16 ***
## X1 0.196884 0.004604 42.76 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.983 on 998 degrees of freedom
## Multiple R-squared: 0.6469, Adjusted R-squared: 0.6466
## F-statistic: 1829 on 1 and 998 DF, p-value: < 2.2e-16m2 <- lm(data = df, formula = Y ~ X2)
summary(m2)##
## Call:
## lm(formula = Y ~ X2, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.7243 -2.8832 0.0466 3.1809 13.0697
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.38628 0.22470 81.83 <2e-16 ***
## X2 -0.64844 0.03535 -18.34 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.342 on 998 degrees of freedom
## Multiple R-squared: 0.2521, Adjusted R-squared: 0.2514
## F-statistic: 336.5 on 1 and 998 DF, p-value: < 2.2e-16m3 <- lm(data = df, formula = Y ~ X3)
summary(m3)##
## Call:
## lm(formula = Y ~ X3, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.5165 -3.5764 0.1243 3.5487 15.4947
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.218702 0.296048 51.41 <2e-16 ***
## X3 -0.004098 0.010778 -0.38 0.704
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.021 on 998 degrees of freedom
## Multiple R-squared: 0.0001449, Adjusted R-squared: -0.000857
## F-statistic: 0.1446 on 1 and 998 DF, p-value: 0.7038m <- lm(data = df, formula = Y ~ X1 + X2 + X3)
coef(m)## (Intercept) X1 X2 X3
## 9.68859064 0.18696066 -0.25223185 -0.03090319summary(m)##
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.5103 -1.7952 -0.0694 1.8003 7.7340
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.688591 0.261698 37.022 < 2e-16 ***
## X1 0.186961 0.005611 33.321 < 2e-16 ***
## X2 -0.252232 0.036261 -6.956 6.33e-12 ***
## X3 -0.030903 0.009354 -3.304 0.000987 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.69 on 996 degrees of freedom
## Multiple R-squared: 0.7135, Adjusted R-squared: 0.7126
## F-statistic: 826.9 on 3 and 996 DF, p-value: < 2.2e-16plot(fitted(m), residuals(m))
hist(residuals(m))
qqnorm(residuals(m))
f <- (summary(m)$r.squared * (nrow(df) - (ncol(df) - 1) - 1))/((1 - summary(m)$r.squared) *
(ncol(df) - 1))
f## [1] 826.8652Challenge 2
library(curl)
f <- curl("https://raw.githubusercontent.com/fuzzyatelin/fuzzyatelin.github.io/master/AN588_Fall21/zombies.csv")
z <- read.csv(f, header = TRUE, sep = ",", stringsAsFactors = TRUE)
head(z)## id first_name last_name gender height weight zombies_killed
## 1 1 Sarah Little Female 62.88951 132.0872 2
## 2 2 Mark Duncan Male 67.80277 146.3753 5
## 3 3 Brandon Perez Male 72.12908 152.9370 1
## 4 4 Roger Coleman Male 66.78484 129.7418 5
## 5 5 Tammy Powell Female 64.71832 132.4265 4
## 6 6 Anthony Green Male 71.24326 152.5246 1
## years_of_education major age
## 1 1 medicine/nursing 17.64275
## 2 3 criminal justice administration 22.58951
## 3 1 education 21.91276
## 4 6 energy studies 18.19058
## 5 3 logistics 21.10399
## 6 4 energy studies 21.48355m <- lm(data = z, height ~ weight + age)
summary(m)##
## Call:
## lm(formula = height ~ weight + age, data = z)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2278 -1.1782 -0.0574 1.1566 5.4117
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.763388 0.470797 67.47 <2e-16 ***
## weight 0.163107 0.002976 54.80 <2e-16 ***
## age 0.618270 0.018471 33.47 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.64 on 997 degrees of freedom
## Multiple R-squared: 0.8555, Adjusted R-squared: 0.8553
## F-statistic: 2952 on 2 and 997 DF, p-value: < 2.2e-16ANCOVA
library(car)## Loading required package: carDatam <- lm(data = z, formula = height ~ gender + age)
summary(m)##
## Call:
## lm(formula = height ~ gender + age, data = z)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.1909 -1.7173 0.1217 1.7670 7.6746
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.74251 0.56869 82.19 <2e-16 ***
## genderMale 4.00224 0.16461 24.31 <2e-16 ***
## age 0.94091 0.02777 33.88 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.603 on 997 degrees of freedom
## Multiple R-squared: 0.6361, Adjusted R-squared: 0.6354
## F-statistic: 871.5 on 2 and 997 DF, p-value: < 2.2e-16m.aov <- Anova(m, type = "II")
m.aov## Anova Table (Type II tests)
##
## Response: height
## Sum Sq Df F value Pr(>F)
## gender 4003.9 1 591.15 < 2.2e-16 ***
## age 7775.6 1 1148.01 < 2.2e-16 ***
## Residuals 6752.7 997
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1plot(fitted(m), residuals(m))
hist(residuals(m))
qqnorm(residuals(m))
library(ggplot2)
p <- ggplot(data = z, aes(x = age, y = height)) + geom_point(aes(color = factor(gender))) +
scale_color_manual(values = c("goldenrod", "blue"))
p <- p + geom_abline(slope = m$coefficients[3], intercept = m$coefficients[1],
color = "goldenrod4")
p <- p + geom_abline(slope = m$coefficients[3], intercept = m$coefficients[1] +
m$coefficients[2], color = "darkblue")
p
CI and Prediction
m <- lm(data = z, formula = height ~ age + gender)
summary(m)##
## Call:
## lm(formula = height ~ age + gender, data = z)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.1909 -1.7173 0.1217 1.7670 7.6746
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.74251 0.56869 82.19 <2e-16 ***
## age 0.94091 0.02777 33.88 <2e-16 ***
## genderMale 4.00224 0.16461 24.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.603 on 997 degrees of freedom
## Multiple R-squared: 0.6361, Adjusted R-squared: 0.6354
## F-statistic: 871.5 on 2 and 997 DF, p-value: < 2.2e-16confint(m, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 45.6265330 47.8584809
## age 0.8864191 0.9954081
## genderMale 3.6792172 4.3252593Challenge 3
ci <- predict(m, newdata = data.frame(age = 29, gender = "Male"), interval = "confidence",
level = 0.95)
ci## fit lwr upr
## 1 78.03124 77.49345 78.56903pi <- predict(m, newdata = data.frame(age = 29, gender = "Male"), interval = "prediction",
level = 0.95)
pi## fit lwr upr
## 1 78.03124 72.89597 83.16651Interactions between Predictions
m <- lm(data = z, height ~ age + gender + age:gender) # or
summary(m)##
## Call:
## lm(formula = height ~ age + gender + age:gender, data = z)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7985 -1.6973 0.1189 1.7662 7.9473
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 48.18107 0.79839 60.348 <2e-16 ***
## age 0.86913 0.03941 22.053 <2e-16 ***
## genderMale 1.15975 1.12247 1.033 0.3018
## age:genderMale 0.14179 0.05539 2.560 0.0106 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.595 on 996 degrees of freedom
## Multiple R-squared: 0.6385, Adjusted R-squared: 0.6374
## F-statistic: 586.4 on 3 and 996 DF, p-value: < 2.2e-16m <- lm(data = z, height ~ age * gender)
summary(m)##
## Call:
## lm(formula = height ~ age * gender, data = z)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7985 -1.6973 0.1189 1.7662 7.9473
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 48.18107 0.79839 60.348 <2e-16 ***
## age 0.86913 0.03941 22.053 <2e-16 ***
## genderMale 1.15975 1.12247 1.033 0.3018
## age:genderMale 0.14179 0.05539 2.560 0.0106 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.595 on 996 degrees of freedom
## Multiple R-squared: 0.6385, Adjusted R-squared: 0.6374
## F-statistic: 586.4 on 3 and 996 DF, p-value: < 2.2e-16coefficients(m)## (Intercept) age genderMale age:genderMale
## 48.1810741 0.8691284 1.1597481 0.1417928library(ggplot2)
library(gridExtra)
p1 <- ggplot(data = z, aes(x = age, y = height)) + geom_point(aes(color = factor(gender))) +
scale_color_manual(values = c("goldenrod", "blue"))
p1 <- p1 + geom_abline(slope = m$coefficients[2], intercept = m$coefficients[1],
color = "goldenrod4")
p1 <- p1 + geom_abline(slope = m$coefficients[2] + m$coefficients[4], intercept = m$coefficients[1] +
m$coefficients[3], color = "darkblue")
p1
p2 <- ggplot(data = z, aes(x = age, y = height)) + geom_point(aes(color = factor(gender))) +
scale_color_manual(values = c("goldenrod", "blue")) + geom_smooth(method = "lm",
aes(color = factor(gender), fullrange = TRUE))## Warning in geom_smooth(method = "lm", aes(color = factor(gender), fullrange =
## TRUE)): Ignoring unknown aesthetics: fullrangegrid.arrange(p1, p2, ncol = 2)## `geom_smooth()` using formula = 'y ~ x'
Challenge 4
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:car':
##
## recode## The following object is masked from 'package:gridExtra':
##
## combine## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionf <- curl("https://raw.githubusercontent.com/fuzzyatelin/fuzzyatelin.github.io/master/AN588_Fall21/KamilarAndCooperData.csv")
d <- read.csv(f, header = TRUE, sep = ",", stringsAsFactors = TRUE)
head(d)## Scientific_Name Family Genus Species
## 1 Allenopithecus_nigroviridis Cercopithecidae Allenopithecus nigroviridis
## 2 Allocebus_trichotis Cercopithecidae Allocebus trichotis
## 3 Alouatta_belzebul Atelidae Alouatta belzebul
## 4 Alouatta_caraya Atelidae Alouatta caraya
## 5 Alouatta_guariba Atelidae Alouatta guariba
## 6 Alouatta_palliata Atelidae Alouatta palliata
## Brain_Size_Species_Mean Brain_Size_Female_Mean Brain_size_Ref
## 1 58.02 53.70 Isler et al 2008
## 2 NA NA
## 3 52.84 51.19 Isler et al 2008
## 4 52.63 47.80 Isler et al 2008
## 5 51.70 49.08 Isler et al 2008
## 6 49.88 48.04 Isler et al 2008
## Body_mass_male_mean Body_mass_female_mean Mass_Dimorphism
## 1 6130 3180 1.928
## 2 92 84 1.095
## 3 7270 5520 1.317
## 4 6525 4240 1.539
## 5 5800 4550 1.275
## 6 7150 5350 1.336
## Mass_Ref MeanGroupSize AdultMales AdultFemale AdultSexRatio
## 1 Isler et al 2008 NA NA NA NA
## 2 Smith and Jungers 1997 1.00 1.00 1.0 NA
## 3 Isler et al 2008 7.00 1.00 1.0 1.00
## 4 Isler et al 2008 8.00 2.30 3.3 1.43
## 5 Isler et al 2008 6.53 1.37 2.2 1.61
## 6 Isler et al 2008 12.00 2.90 6.3 2.17
## Social_Organization_Ref
## 1
## 2 Kappeler 1997
## 3 Campbell et al 2007
## 4 van Schaik et al. 1999; Kappeler and Pereira 2003; Nunn & van Schaik 2000
## 5 Campbell et al 2007
## 6 van Schaik et al. 1999; Kappeler and Pereira 2003; Nunn & van Schaik 2000
## InterbirthInterval_d Gestation WeaningAge_d MaxLongevity_m LitterSz
## 1 NA NA 106.15 276.0 1.01
## 2 NA NA NA NA 1.00
## 3 NA NA NA NA NA
## 4 337.62 187 323.16 243.6 1.01
## 5 NA NA NA NA NA
## 6 684.37 186 495.60 300.0 1.02
## Life_History_Ref GR_MidRangeLat_dd Precip_Mean_mm Temp_Mean_degC AET_Mean_mm
## 1 Jones et al. 2009 -0.17 1574.0 25.2 1517.8
## 2 -16.59 1902.3 20.3 1388.2
## 3 -6.80 1643.5 24.9 1286.6
## 4 Jones et al. 2009 -20.34 1166.4 22.9 1193.1
## 5 -21.13 1332.3 19.6 1225.7
## 6 Jones et al. 2009 6.95 1852.6 23.7 1300.0
## PET_Mean_mm Climate_Ref HomeRange_km2 HomeRangeRef DayLength_km
## 1 1589.4 Jones et al. 2009 NA NA
## 2 1653.7 Jones et al. 2009 NA NA
## 3 1549.8 Jones et al. 2009 NA NA
## 4 1404.9 Jones et al. 2009 NA 0.40
## 5 1332.2 Jones et al. 2009 0.03 Jones et al. 2009 NA
## 6 1633.9 Jones et al. 2009 0.19 Jones et al. 2009 0.32
## DayLengthRef Territoriality Fruit Leaves Fauna DietRef1
## 1 NA NA
## 2 NA NA
## 3 NA 57.3 19.1 0.0 Campbell et al. 2007
## 4 Nunn et al. 2003 NA 23.8 67.7 0.0 Campbell et al. 2007
## 5 NA 5.2 73.0 0.0 Campbell et al. 2007
## 6 Nunn et al. 2003 0.6506 33.1 56.4 0.0 Campbell et al. 2007
## Canine_Dimorphism Canine_Dimorphism_Ref Feed Move Rest Social
## 1 2.210 Plavcan & Ruff 2008 NA NA NA NA
## 2 NA NA NA NA NA
## 3 1.811 Plavcan & Ruff 2008 13.75 18.75 57.30 10.00
## 4 1.542 Plavcan & Ruff 2008 15.90 17.60 61.60 4.90
## 5 1.783 Plavcan & Ruff 2008 18.33 14.33 64.37 3.00
## 6 1.703 Plavcan & Ruff 2008 17.94 12.32 66.14 3.64
## Activity_Budget_Ref
## 1
## 2
## 3 Campbell et al. 2007
## 4 Campbell et al. 2007
## 5 Campbell et al. 2007
## 6 Campbell et al. 2007d <- select(d, Brain_Size_Female_Mean, Family, Body_mass_female_mean, MeanGroupSize,
DayLength_km, HomeRange_km2, Move)m <- lm(data = d, log(HomeRange_km2) ~ log(Body_mass_female_mean) + log(Brain_Size_Female_Mean) +
MeanGroupSize + Move)
summary(m)##
## Call:
## lm(formula = log(HomeRange_km2) ~ log(Body_mass_female_mean) +
## log(Brain_Size_Female_Mean) + MeanGroupSize + Move, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.7978 -0.6473 -0.0038 0.8807 2.1598
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.955853 1.957472 -3.553 0.000865 ***
## log(Body_mass_female_mean) 0.315276 0.468439 0.673 0.504153
## log(Brain_Size_Female_Mean) 0.614460 0.591100 1.040 0.303771
## MeanGroupSize 0.034026 0.009793 3.475 0.001095 **
## Move 0.025916 0.019559 1.325 0.191441
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.127 on 48 degrees of freedom
## (160 observations deleted due to missingness)
## Multiple R-squared: 0.6359, Adjusted R-squared: 0.6055
## F-statistic: 20.95 on 4 and 48 DF, p-value: 4.806e-10plot(m$residuals)
qqnorm(m$residuals)
shapiro.test(m$residuals)##
## Shapiro-Wilk normality test
##
## data: m$residuals
## W = 0.96517, p-value = 0.1242m <- lm(data = d, log(HomeRange_km2) ~ log(Body_mass_female_mean) + log(Brain_Size_Female_Mean) +
MeanGroupSize)
summary(m)##
## Call:
## lm(formula = log(HomeRange_km2) ~ log(Body_mass_female_mean) +
## log(Brain_Size_Female_Mean) + MeanGroupSize, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.7982 -0.7310 -0.0140 0.8386 3.1926
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.354845 1.176046 -4.553 1.45e-05 ***
## log(Body_mass_female_mean) -0.181627 0.311382 -0.583 0.560972
## log(Brain_Size_Female_Mean) 1.390536 0.398840 3.486 0.000721 ***
## MeanGroupSize 0.030433 0.008427 3.611 0.000473 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.228 on 103 degrees of freedom
## (106 observations deleted due to missingness)
## Multiple R-squared: 0.6766, Adjusted R-squared: 0.6672
## F-statistic: 71.85 on 3 and 103 DF, p-value: < 2.2e-16plot(m$residuals)
qqnorm(m$residuals)
shapiro.test(m$residuals) # no significant deviation from normal##
## Shapiro-Wilk normality test
##
## data: m$residuals
## W = 0.99366, p-value = 0.9058